Integral Formula Research Articles

New proofs of theorems on q-orthogonal functions

In this paper, we establish a [Formula: see text]-integral formula by using the orthogonality relation, and also provide a new proof of the [Formula: see text]-orthogonality relation for the continuous [Formula: see text]-ultraspherical polynomials. A new [Formula: see text]-beta integral with five parameters is evaluated.

Product formulas for multiple stochastic integrals associated with Lévy processes

AbstractIn the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

On the Laplace equation on bounded subanalytic manifolds

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, possibly non closed. We also establish density results for W∇1,p(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{W}}^{1,p}_\ abla (M)$$\\end{document}, M bounded subanalytic manifold, which is the space of the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} tangent vector fields v on M for which ∇v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla v$$\\end{document} is Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, where ∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla $$\\end{document} is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the p-Laplace equation, for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,\\infty )$$\\end{document} large.

Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.

On the Nörlund-Rice Integral Formula

After introducing the famous Nörlund-Rice integral formula, we apply it to Laguerre polynomials, Melzak’s relation, and Stirling numbers of the second kind to obtain nice expressions.

Aeroelastic Structural Analysis to Calculate Symmetrical Divergence Modes of an Aircraft Wing Using Aerodynamic Lifting Line Theory

This work explores the use of numerical matrix iteration to determine an aircraft wing's divergence speed using the aerodynamic span effects approach. The present work begins with the construction of equilibrium equations in differential and integral forms. In this paper, integral formulas are used because it serves as a convenient basis for numerical solutions of complex practical problems. Second, the straight-tapered wing is divided into a number of Multhopp stations. Subsequently, the stations' torsional influence coefficient matrix has been calculated. Third, lifting line theory was used as a suitable choice of aerodynamic theory, and the governing equations were represented in matrix form. Finally, aerodynamic span effects are taken into consideration with induction effects according to Prandtl’s lifting line theory to calculate the symmetrical divergence speed from the lowest eigenvalue of the homogenous governing integral equation. To get the solution to converge, a matrix has been iterated using the MATLAB environment. The obtained results will aid modern aircraft designers in their understanding of wing instability in steady motion.

Deductions, Applications and Expansion of the Cauchy’s Residue Theorem

Cauchy’s integral formula is one of the most important discovery in the development history of the complex variable integration. This article focuses on the methods of the integration in mathematics. The author aims to write about how to deduce the residue theorem from Cauchy’s integral formula and how to use the Cauchy’s integral theorem and residue theorem in the integral questions. The following parts of the article include using method of limits and Euler’s formula to deduce roughly the Cauchy’s Integral formula. In addition, the author also uses the differentiation to transfer the integral function. In part three, the author calculates three different types of examples and considers that residue theorem and Cauchy’s integral formula are very useful in the contour integral with singularities. The article is in order to show that the practicability of these theorem and formula. These formulas and theorems are used in a wide range of the subjects and areas.

Residue Theorem and Logarithmic Residue Theorem

This passage is written for researching the application of residue theorem and Logarithmic Residue theorem. The residue theorem is a powerful tool for the analysis of complex functions. It also can be used to calculate the integral of the real functions. Residue theorem is the extension of the Cauchy’s integral theorem and Cauchy integral formula. Passage researched two theorems by defining the residue in two ways. Each of them is defined by Laurent series and defined by integral. Then the residue theorem is introduced with its definition and applications. The passage points out two instances for the applications of the residue theorem. After the residue theorem, the Logarithmic Residue theorem is listed with its definition and applications. There are also three instances for its applications. The residue theorem and Logarithmic Residue theorem mainly contribute for the research of the complex functions’ integral calculations. The residue theorem provides powerful mathematical tools in certain special types of real integration problems.

Integral Representation Formulas in Banach Algebra Settings

Integral Representation Formulas in Banach Algebra Settings

GNSS-assisted optimal alignment method for low-cost SINS motion of vehicle

Abstract To improve the alignment accuracy and environmental applicability of the micro-electromechanical system (MEMS)-based strapdown inertial navigation system (SINS), this study proposes a global navigation satellite system (GNSS)-assisted multi-vector determination of the attitude optimal indirect coarse alignment. Using the inertial information output from the inertial measurement unit and the velocity information from the GNSS, a simplified velocity observation vector is constructed and the velocity lever arm stemming from the mounting position inconsistencies of the GNSS and SINS is fed back into the velocity of the carrier system. When constructing the observation vectors, the integration interval is shortened by reconstructing the two-vector integration formula for reducing the cumulative error of the inertial device. The attitude matrix problem is defined as the Wahba problem, which is solved using the singular value decomposition method. Based on the relationship between gyro zero bias and misalignment angle, the corresponding state and measurement equations are designed. Furthermore, owing to the measurement noise uncertainty, the adaptive traceless Kalman filtering algorithm is introduced to realize the effective adaptation processing of the measurement noise. More accurate attitude matrix estimates are obtained by continuously correcting the carrier system transformation matrix. The running car experiment results show that the proposed method exhibits higher alignment accuracy and environmental applicability than the current MEMS strapdown inertial navigation coarse alignment method and the traditional optimization-based alignment method.

Lorentz invariants in particle-wave mechanical systems

In these notes we detail a number of new results involving the Lorentz invariants associated with the special relativistic extension of Newton’s second law proposed in [10] and not included in that text. We first summarise existing results for the two Lorentz invariants ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} and η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} and the angle θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} which is the angle in which Lorentz invariance appears as a translational invariance. We then determine new integral formulae involving these quantities and a new relationship which connects the applied forces with the same invariances. This latter relation turns out to be equivalent to the partial differential equation arising from the invariance of the velocity equation dx/dt=u(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}x/\ extrm{d}t = u(x, t)$$\\end{document} under a Lorentz transformation. We then provide some specific examples which confirm the validity of the new relation connecting the applied forces with the invariances.

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Extension of Pochhammer symbol, generalized hypergeometric function and τ-Gauss hypergeometric function

Abstract We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented.

Two methods based on spline quasi-interpolants to estimate volumes enclosed by parametric surfaces

In this study, we explore two methods based on spline quasi−interpolants (abbr. QI) that rely exclusively on point evaluations to estimate the volume of a closed parametric surface. The first method involves a division of this surface evenly into parallel slices in order to approximate the contours of each slice using quadratic QI. Subsequently, we calculate numerically the areas of the interior surfaces, which will be used in a quadrature formula to obtain the approximate volume. The second method uses a bivariate QI to calculate the entire area, and an integration formula is applied to estimate the desired volume. In both methods, we study the associated error estimates. Numerical examples are provided to illustrate our methods, as well as a real-world application, as well as a study of the effect of noise on the robustness of both volume estimation methods.

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

Meshless Technique Based on Moving Least Squares Approximation for Numerical Solutions of Linear and Nonlinear Third-Kind VIEs

The principal objective of this paper is to propose a numerical scheme to solve linear and nonlinear third-kind Volterra integral equations (VIEs). The method approximates the solution using the collocation method based on moving least squares (MLS) approximation. The suggested scheme is meshless, since it needs no background mesh or cell structures, and is thus independent of the geometry of the domain. The approach reduces the solution of third-kind VIEs to the solution of systems of algebraic equations. All integrals appearing in the approach are calculated approximately by the Gauss–Legendre integration formula. The convergence analysis of the proposed scheme has been conducted, and its performance has been tested through a series of numerical examples, demonstrating its effectiveness and applicability in comparison to the existing methods.

Explicit bivariate simplicial depth

The simplicial depth (SD) is a celebrated tool defining elements of nonparametric and robust statistics for multivariate data. While many properties of SD are well-established, and its applications are abundant, explicit expressions for SD are known only for a handful of the simplest multivariate probability distributions. This paper deals with SD in the plane. It (i) develops a one-dimensional integral formula for SD of any properly continuous probability distribution, (ii) gives exact explicit expressions for SD of uniform distributions on (both convex and non-convex) polygons in the plane or on the boundaries of such polygons, and (iii) discusses several implications of these findings to probability and statistics: (a) An upper bound on the maximum SD in the plane, (b) an implication for a test of symmetry of a bivariate distribution, and (c) a connection of SD with the classical Sylvester problem from geometric probability.

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

(Two-scale) W1LΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1}L^{\\Phi }$$\\end{document}-gradient Young measures and homogenization of integral functionals in Orlicz–Sobolev spaces

(Two-scale) gradient Young measures in Orlicz–Sobolev setting are introduced and characterized providing also an integral representation formula for non convex energies arising in homogenization problems with nonstandard growth.